truth functional connectives

8/7/2019 Truth Functional Connectives

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Consider the Followingstatements:

(S1) It is not true that he issick.

(S2) Plato is a philosopher and Plato is a Greek.

(S3) She dances gracefullyor she sings very well.(S4) If you get a hundred on

every exam, then I willtreat you to the moviehouse.

(S5) I will earn my collegedegree if and only i f Iwill pass all myacademic requirements.

T he highlighted word/wordsare called truth functionalconnectives

Why connectives ?Because they connect simplestatements and form acompound statement.

Why T ruth Functional?they determine the truthvalue of statements.truth and f alsity as truthvalues , which arerespectively abbreviated as Tand F.


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K inds o f Truth FunctionalConnectives


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1. NEGATION ( ~ )T his one-place connective is called

negation, and is symbolized by~ (tilde), which is a stylized

form of n, short for negation.It is not true that_______________

Example: It is not true that heis sick.

T his can be symbolized as: ~S (Sis an arbitrary representation of the statement, He is sick. )

A ~ A S ~ S

TF

TF

F T F T

We can formalize the logicalargument in the form of a

TRUTH TABLE.TABLE 1 S1


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2 . CONJUNCTION (& )T wo-place connective

___________and ___________ Example: Plato is a philosopher

and Plato is a Greek.

T his can be symbolized as: P & G

P for Plato is a philosopherG for Plato is a Greek& f or and

So we got: P & G

T ABLE 2 S2

A B A & B P G P & G

T T T T T T

T F F T F F

F T F F T F

F F F F F F

Rule: A conjunction A&B is true, if bothconjuncts are true, otherwise, it is false


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3 . DISJUNCTION ( )T wo-place connective

____________ or______________

Example: She dances gracefully or she sings very well.

T his can be symbolized as: D S

D for She dances gracefullyS for She sings very wellv for or

So we got: D S

In English, the word or has at leasttwo different meanings, or senses, which are respectivelycalled the exclusive sense and

the inclusive sense.

She will go to the party or shewill stay home.

She dances gracefully or shesings very well.


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T ABLE 3 S3

A B A B D S D S

T T T T T T

T F T T F T

F T T F T T

F F F F F F

Rule:Disjunction is false whenever the twodisjuncts are false

3 . DISJUNCTION ( )

Example: She dances gracefully or she sings very well.

T his can be symbolized as: D S

D for She dances gracefullyS for She sings very wellv for or

So we got: D S


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T ABLE 4 S4

A B A B H T H T

T T T T T T

T F F T F F

F T T F T T

F F T F F T

Rule: A conditional A B is false if theantecedent (A) is true and theconsequent (B) is false; otherwise, it is

true.

4 . CONDITIONAL ( )T wo-place connective

If___________,then_______________ antecedent consequent

Example: If you get a hundred onevery exam, then I willtreat you to the moviehouse.

T his can be symbolized as: H T

H for If you get a hundred on everyexamT for I will treat you to the movie

housefor ifthen

So we got: H T


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T ABLE 4 S5

A B A B D P D P

T T T T T T

T F F T F F

F T F F T F

F F T F F T

Rule: A biconditional A B is true if and only if the constituents A, B have the same truthvalue; otherwise it is false.

5 . BICONDITIONAL ( )T wo-place connective

__________ if and only if___________ Example: I will earn my college degree

if and only i f I will pass all myacademic requirements.

T his can be symbolized as: D P

D for I will earn my college degreeP for I will pass all my academic

requirementsfor if and only if

So we got: D P


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Application o f Symbolic Logicin the Electric Circuit


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T he & circuit series

A B A & B Light

T T T O n

T F F O ff

F T F O ff

F F F O ff

T he table shows that only when both switches A and B aredown/true can the light be ON and the statement is true.

Switch is up F/0Switch is down T /1


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The v circuit-parallel

A B A B LightT T T O nT F T O n

F T T O n

F F F O ff

T he table shows that if either switch or both switches are down/true , the light isO n, and the proposition is true . But if both switches are up/false then the lightis O ff and proposition is false.


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Biblical Text AnalysisLast Sunday Gospel

Scene 1: Jesus appeared to some of the disciples

Again Jesus said, "Peace be with you! As the Father has sent me, I am

sending you."22

And with that he breathed on them and said, "Receive theHoly Spirit. 23 If you forgive anyone his sins, they are forgiven; if you do notforgive them, they are not forgiven.

Standard: (if Y, then T ); (If not Y, then not T )Symbolized: (Y T ); (~ Y ~T )

T ranslating to argument the first statement.If you forgive anyone his sins, then they are forgiven. You forgive

anyone his sins. T herefore, they are forgiven.Y T,

Y:./ T


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Biblical Text AnalysisBeatitudes

Blessed are the meek,for they shall inherit the earth.

Blessed are the merciful,for they shall obtain mercy.

Blessed are the peacemakers,for they shall be called children of God.

Blessed are they who are persecuted for the sake of righteousness,for theirs is the kingdom of heaven."

Logical Standard Form

If they are meek, then they will inherit the earth. M I

If they are merciful, then they will obtain mercy. M O

If they are peacemakers, then

they are called children of God.P C

If they are persecuted for the sakeof righteousness, then theirs is thekingdom of heaven." P H


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Analysis o f NewsDESER T -LIKE DAM. T he scorching El N ino heat dries up the Magat Dam,Isabela, making it look like a virtual sand dune. If the dam s water levelcontinues to go down, it is feared that Luzon may be hit by intermitten power outrages. (Mark Balmores; Front Page - Manila Bulletine 4/12/10

Formalized the statements into Standard Form.1). If the Magat dam s water level continues to go down, then it is feared thatLuzon maybe hit by intermitten power outrages. 2) T he scorching El N ino heatdries up the Magat Dam, Isabela and El N ino is making Magat Dam, Isabelalook like a virtual sand dune. 3) T herefore, Magat Dam is like a desert.

Symbolization: If M, then L 2) D and M 3)T

herefore, MSymbolic Language: M L,E&M

:./ M


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EXERC ISES (Set A)Direction: Compute the truth values of the following symbolic statements,

supposing that the truth value of A, B, and C is T , and the truth value of X,Y, Z is F.

Set B __ T ___1. ~ A B _____2. Av ~ Y _____3. B& ~ C _____4. ~ Y&Z _____5. A X _____6. B Y _____7. Y Z _____8. A B _____9. X (Y Z) _____10. ~ (C&Y) (A&Z) _____11. A (B Z)

_____12. (A X) & (Y B)

Answer Key1. T2. T3. F4. F5. F

6. F7. T8. T9. F10. F11. F

12. T


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EXCERC ISE

Set B ( Write your answer in one whole yellow pad paper)Direction: Symbolize the following statements using the first

letter of the underlined word and construct its truth table.

1. If it is brown out then the computer will shutdown2. Either I will eat ice cream or pancake.3. Honda and Zuzuki build motorcycles.4. T he shortage of water decreases if and if Elni o will

stop.5. It is not true that all Filipinos are poor.

truth functional connectives

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